There's no key binding in the controls menu, and there's no key binding mentioned in the manual. I've tried purposefully getting damaged, then hitting every key on my keyboard, and it doesn't activate. I own it, and it is equipped on my ship, but it almost seems like they forgot to make them usable. Has anyone figured this out?

The repair kits are upgrades you install onto the ship, not consumables. They're always functioning at a fixed rate, thing is, you might've missed the fact that the first one or two only repair the 'internal components' (I know I did back in the day) so they won't actually recover your armor at all. They're still good to have, but you'll have to progress further into the story to be able to buy the models that actually heal you - and even those aren't a miracle, but on longer missions they're quite useful nonetheless.